Bring a large pot of water to a boil. Prepare a bowl of ice water. Add the Brussels sprouts and boil until they start to get tender, 5-7 minutes. Drain and place the sprouts in the ice water to stop the cooking. Once cooled, drain and dry the sprouts. Cut the sprouts in half through the root, or into quarters if they are especially large.

Put 2 tablespoons of butter in a skillet and melt over medium heat. Whisk in the flour and cook for a couple minutes. Slowly add the milk, whisking to avoid any lumps. Cook until thickened. Season to taste with salt, pepper and freshly grated nutmeg. Add the Parmesan cheese and stir until melted.

Place the Brussels sprouts in a 2-quart baking dish. Pour the sauce evenly over the sprouts. Sprinkle the breadcrumbs over the top. Cut the remaining 1 tablespoon of butter into small pieces and sprinkle over the top of the breadcrumbs.

Bake in the preheated oven until the sauce is hot and bubbly, about 20 minutes. Turn on the broiler and broil until the breadcrumbs are toasted, 1-2 minutes. Serve.